Consider the i.i.d. Gaussian variables , where and . The random variables from a linear combination of , e.g.
are correlated. Their covariance is given by
as the covariance between two independent standard normal variables is zero.
Consider a -vector of independent standard normal variables , and define the -vector of random variables as the linear combination of the variables in and a -vector of constants :
then the covariance matrix of is and the mean is .
The p.d.f. of a univariate Gaussian variable is
while the p.d.f. of a multivariate Gaussian variable (-vector) is